Int. J. EngngSci. Vol. 32, No. 3. pp. 481-500, 1994
Copyright@ 1994 Elsevicr ScienceLtd
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Petgamoo
MODELING ELECTRO-RHEOLOGICAL
MATERIALS
THROUGH MIXTURE THEORY
K. R. RAJAGOPAL
and R. C. YALAMANCHILI
Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A.
A. S. WINEMAN
Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor,
MI48109, U.S.A.
(Communicated by C. G. SPEZIALE)
Abstract-Electra-rheological
materials are suspensions of particles in non-conducting fluids, and all
models that have been developed to date describe their behavior by treating them as a homogenized
single continuum, and ignoring the multicomponent structure of the material. The theory of
interacting continua is ideally suited for modeling such mixtures and in this paper we present a simple
theory which takes into account the distribution of the particles in the fluid, the applied electric field,
and the relative motion of the two constituents. To illustrate the utility of such a theory we study the
flow of an electro-rheological material between two parallel plates under the application of an
electrical field normal to the plates.
1.
INTRODUCTION
Electra-rheological
materials are suspensions of non-conducting particulate media in nonconducting fluids. Properties like viscosity of the suspension change significantly on the
application of an electric field. This phenomenon was observed over three decades ago by
Winslow [l]. Such behavior can be gainfully employed in a wide range of technological
applications from the design of clutches, brakes, shock absorbers and journal bearings to a
plethora of applications in hydraulics. Much of the effort that has been expended in recent
years in the field of electro-rheology is in designing and tailor-making these materials. There
has also been a reasonable amount of experimental work on electro-rheological
materials.
However, little if any effort has been directed towards providing a comprehensive theory to
describe the behavior of these materials. Recently, Rajagopal and Wineman [2] developed a
mathematical model for field dependent materials based on the basic principles of continuum
mechanics, which predicts behavior in keeping with experimentally observed phenomena. The
theory of Rajagopal and Wineman [2] assumes that the electro-rheological suspension can be
regarded as a single continuum. A good case can be made for such an approach on the basis of
homogenization which yields average properties for the suspensions. However, it would be
remiss not to try to model such a suspension via the theory of interacting continua (cf.
Truesdell [3,4]). I n such an approach, balance laws are postulated for each constituent which
allows for interaction between the constituents including the possibility of generation of the
individual species, chemical reactions, electro-mechanical,
electro-chemical and other effects.
The theory also allows us to account for the fact that we have particles moving through the
fluid by including interactions such as drag, virtual mass effect, magnus effect, spin-lift, density
gradient effects, buoyancy effects amongst others.
Mixtures of fluids and solid particulate media within the context of a purely mechanical
theory of interacting continua have been studied by Massoudi [5], and Johnson et al. [6, 71.
These studies were primarily aimed at problems of fluidization and the transport of mixtures of
fluids and granular solids. The mixture was assumed to be made up of a classical linearly
viscous fluid and a granular solid. Thus, the partial stresses associated with the fluid and the
granular material depend on the stretching tensors associated with both the fluid and solid
motion in addition to the way in which the solid is distributed, which is described by a volume
distribution function u and the gradient of the volume distribution function.
ES 32:3-6
481
K. R. RAJAGOPAL
482
et al.
Here, we shall assume that the partial stresses associated with the fluid and the granular solid
also depend on the electrical field E. As a consequence of this, we find that there are
contributions to the normal stress differences due to the electrical field E, the gradient of the
volume fraction, grad u and also the squares of the stretching tensors Dz and D:, for the solid
and fluid respectively.
An interesting consequence of our study is that an appropriately defined mixture viscosity
changes with position, due to changes in the velocity gradients of the fluid and granular solid,
though the individual viscosities of the fluid and granular solid are constant. We find that our
mixture shear thickens.
The outline of the paper is as follows. In Section 2 we introduce the basic equations of
mixture theory. In Section 3 we introduce the constitutive equations for electro-rheological
materials. In the final section we discuss the flow of electro-rheological
materials between
parallel plates, namely plane Couette and Poiseuille flow.
2. PRELIMINARIES
AND
BASIC
EQUATIONS
OF MIXTURE
THEORY
For the sake of completeness and continuity we shall provide a very concise review of the
basic equations of mixture theory. A detailed and up-to-date account of the same can be found
in the several appendices in [4].
The basic assumption of mixture theory is that each point in the domain of the mixture is
occupied by a particle belonging to each constituent. That is, we assume that there is a
“homogenized” equivalent continuum corresponding to each constituent, and these “homogenized” continua coexist. Let x” and X’ denote the position of a solid particle and fluid particle,
respectively, in their reference states. The motion of the solid (granular) continuum and the
fluid are then given by invertible one-to-one maps
xs= f(xs,
t),
xf=Xf(X’, t).
(1)
The various kinematical terms associated with the two constituents
can then be defined as
velocity
acceleration
L” = grad ub,
velocity gradient
stretching
spin
D” = ; (L” + (L”)T),
W” =;
(L” -
(L”)T),
L’ = grad uf
D’= ; (L’ + (Lf)T)
w’=
$(L’
-
p/)7‘).
Let ps and of denote the densities of the two constituents
conservation of mass for the constituents takes the form
T
+ div(p”u”) = 0,
z
+ div(p’u’) = 0.
We assume there exists a partial stress tensor associated
(2)
in the mixed state.
The
(4)
with each constituent.
Then the
Modeling electro-rheological materials
balance of linear momentum for the two constituents
483
takes the form
divT+b+psb,=psd$,
(5)
(6)
where b denotes the diffusive body forces on one constituent due to the presence of the other,
and b, denotes the external body force field.
The balance of angular momentum takes the form that
T=T”+Tf=(T+Tf)T=TT.
(7)
Thus, while the total stress is symmetric, the individual partial stresses need not be so.
We shall not document the energy equation or the entropy inequality as it is not relevant to
our subsequent discussion.
3. CONSTITUTIVE
EQUATIONS
FOR ELECTRO-RHEOLOGICAL
MATERIALS
We shall assume that the partial stresses for the fluid Tf, and the granular solid T, and the
interactive body force b have the following representation
Tf = T’(Y, grad Y, E, D’, D”),
(8)
T” = T(Y, grad Y, E, D’, D”),
(9)
and
b = b(p”, pf,
us- uf,asf, D”, D’, w” -
Wf),
where
a” = g
I[
+ [grad u”](u” - uf) -
zf + [grad ur](uf - us)
(10)
1,
(11)
and Y denotes the volume fraction of the solid.
The term us - u‘ in the expression for the diffusive body force b incorporates the drag effect,
while aSf accounts for what is commonly referred to as the virtual mass effect which exists in
virtue of the relative acceleration between the solid and fluid particles. The specific form for asf
that has been chosen is frame indifferent. The interactive body force b can also depend on the
relative spin W - W’, which is frame indifferent though the individual spins w” and Wf are not
so. In general the interaction term b can depend on the electric field E. Also, frame
indifferences requires that the constitutive functions are isotropic functions of all the variables
involved. It is worthwhile pointing out that even though the fluid is non-conducting, we have
assumed that the partial stress on the fluid depends on the electric field. The reason for this is
that the partial stress in the granular solid is affected by the electric field and since the fluid and
solid are jointly occupying the domain of the mixture, the stresses in the fluid are affected by
the presence of the granular solid. If t, represents forces on solid particles due to the fluid, an
electric field creates additional forces between particles. The basis for such a force is yet under
investigation and forms the backbone of important research in material science.
To make the problem tractable, we shall make the simplifying assumption that
T’= T’(v, grad Y, E, D’),
(12)
T = T(Y, grad v, E, D”),
(13)
and
b = b(d
-
uf, asf).
(14)
K. R. RAJAGOPAL
484
et
al.
That is, the partial stress in each constituent depend only on the kinematics associated with the
constituents and only the drag and virtual mass effect are important with regard to interaction.
The first assumption is usually referred to as the principle of phase separation and can be
traced back to the work of Adkins [S].
Let
U = grad v.
Then frame-indifference
(13
(cf. Truesdell [9]) requires that
QT'(v,
WGDf)QT=Tftv,QU,QE,QDfQT),
Qb(u”- uf, asf) = b(Q[u” QTS(OJ,E,WQT=~(~~QUQE,QDQT),
for all Q that are orthogonal.
It follows that Tf and T” have the following representation
(161
uf], Qasf),
(17)
(cf. Spencer [lo])
T’=~,~+(Y&CHJ++~E@E+CY~(UC~E++@U)
+ CK,D’+ c~~(D~)~f a7(DfU (8, U + U @ D’U)
+ CY~((D’)*U63 U + U 63 (D’)*U) + +,(DfE QDE + E @ D’E)
+ ~u,,,((d)~E ‘8 E + E @ (D’)*E) + aI ,(DfU C3E + E C3DfU)
(
8)
f ac,,(DfE ‘8 U + U C3D’E) + (u,,((D?*U 8 E + E 63 (Df)‘U)
+ CY,~((D~)*E‘8 U + U 8 (Df)*E) + a,s[((Df)2U ‘23DfE + D’E C3(D’)*U)
-(D’)‘E
@ DfU - D’U 63~(Df)*E],
and
T”=/3,1+/32U@U+&E@E-t_t4(UC3E+EC3U)+j3sDJ
f &(DS)* + P,(D”U CC’
U + U C3DW) + /3R((DS)2UQDU + U C3(D”)‘U)
+ &(D”E ‘8 E + E QDD”E) + P,,,((DS)‘E 63 E + E @ (D”)2E)
+ & ,(DW @ E + E CzaD’U) + #l,,(D”E 63 U + U C3D”E)
+ &((D)*U
QDE + E C?J(D”)2U) -I-&,((DS)2E @ U + U 03 (D”j2E)
+ &[((DS)2U 60 D”E + WE ‘30(D”)2U) - (D”)*E @ D”U - D”U QD(D”)2E]
where the material parameters
(19)
cu,, i = 1, . . . , 15 are functions of the invariants
tr(U 03 U), tr(E 63 E), trD’, tr(D’)“, tr(Df)3, tr(U 60 E), tr(D’U Q9U),
tr(D’E @ E), tr[(Df)2U 63 U],
tr((D’)*E (20E), tr(DE C3U), tr(Df)2E 8 U,
and v, and the material parameters
(20)
pi, i = 1, . . . , 15 are functions of the invariants
tr(U C3U), tr(E 8 E), trDs, tr(DS)2, tr(D’)“, tr(U @ E), tr(D”U @ U),
tr(D”E QDE), tr[(DS)2U 63 U],
tr[(D”)2E @ E], tr(D”E C3U), tr[(D8)2E @ U],
(21)
and v.
The constitutive expressions (18) and (19) have to be substituted respectively into (5) and (6)
to obtain the equations of motion for the fluid and solid constituents. It is, of course, essential
to assume a specific constitutive expression for the interaction force b which incorporates the
effects of drag, virtual mass, buoyancy, lift and other forces which appear due to the presence
of a second constituent (cf. Johnson et al. 1111). A good starting point is the inclusion of the
effects due to drag and virtual mass, i.e.
b = y,(u” - u’) + y2a*‘,
where y, and y2 are in general functions of v.
(22)
Modeling electro-rheological materials
485
It is possible that the constitutive relations (12) and (13) are simpler than those envisaged.
For example, it might be likely that the stress in the fluid is insensitive to the gradients in the
volume fraction of the granular solid. It is also, at this juncture, unclear how the electric field
affects the stress in the fluid and solid. It is possible that the electric field may not have
significant effects on both the stresses. On the other hand, it might, and we would also have to
include it in the constitutive expression for the interactive force b.
We note that the total stress (or the mixture stress) is defined through T = T + Tf. Then
we need the coefficients as, /3’ and y” to tend to the correct limits as v-0,
namely T’. It is
customary in the two phase flow literature to express the mixture stress T = (1 - v)‘i” + ?. Of
course, our T” and T’ are related to $ and %‘, appropriately. We shall choose to work with the
expressions T’ and T given in (18) and (19), respectively. Also, we are not interested in the
limit v-1
(or v-v,,
v, being the maximum packing). We are in fact interested when
sufficient amount of both the carrier fluid and the particulate media are present in the mixture.
We notice that in order for the theory to be of practical utility, we need to simplify the
constitutive equations drastically. Otherwise, we are faced with a theory that incorporates, in
general, 32 material functions al, . . . , a15, PI, . . . , pls, y, and y2, and even if these material
moduli, which are functions of the various invariants defined in (20) and (21) and possibly Y,
are constants there are way too many of them to be determined in any reasonable experimental
program. This will become evident from the next section where we shall study what is probably
the simplest flow problems, namely plane Couette and Poiseuille flow between parallel plates.
In the final sections we shall discuss some simplifications to the model (18), (19) and (22).
4.
EXAMPLE:
FLOW
BETWEEN
PARALLEL
PLANE
Let us consider the problem of unidirectional steady flow of a mixture modeled by (18), (19)
and (22) between infinite parallel plates (cf. Fig. 1). Suppose that the velocity fields associated
with the fluid and solid, the volume fraction, and the electrical field have the form
uf = uf(y )i,
It follows from (ll),
us = u,( y )i,
E = Ej.
y = Y(Y),
(23)
(18), (19), and (22) that
+u;v,EhI
+
2
a121
}+(j8j)(a,+(Ul)2e2+E2a3+2Ev’ar+~
+ $u;)~(v’)~L~x + (u;)~E~ y
+ (u;)~v’E
(a,3 + a,4)
4
I
+ (k@kb,,
(24)
~=(iDi)(~,+(~~)~~]+[(i~j)+(j~i)](U~~+~~(~’)~~+(U:)E~~
+ u’v’E(B,,
s
+
2
812)
]+(j@j)[~,+(v’)2/?2+E2&+2Ev’~,+~
B
+ (u;)“( v’)~ $ + (u;)‘E2 $! + (u;)‘v’E
(PI3 + PM)
4
) + (k @ k)P,
b = y,(u, - uf)i.
,
(26)
The constitutive expressions for the partial Cauchy stresses T’ and T” reduce to the model of
Johnson et al. [5] for a mixture of a solid infused with particles, when the electrical field E = 0.
Even in this simplified case, and when the material coefficients are constants the balance of
486
K. R. RAJAGOPAL
et al.
-h
(I>)
U
h
*
Fig. 1. (a) Coordinate
system with top plate stationary.
Shaded region is the plug region of constant
velocity; (b) coordinate
system with top plate moving. Shaded region is the plug region of rest.
linear momenta are a system of coupled highly non-linear ordinary differential equations which
have to be solved numerically (cf. Johnson et al. [5]). Here, we are more interested in
recognizing some simple features associated with the flow due to the presence of the electric
field.
The total stress T [cf. (7)] is given by
T = (i 8 i)( (a, + PI) + :(u:)~ + a6 + (~6)‘: &,] + [(i @j) + (,j C3 i)]{ u; 7 + u6 $
+; ~‘WW
+ uhP7) +; m4)~9
+ (j @,j){(a,+ Bd + @(a2
+
P2)
+
(4)P91+; E~Td(~II + a,*) + 4(B,, + ml}
+
E2(a3
+
B3)
+
ZEv’(h
+
P4)
+
(27)
$cU:)‘a6
~~‘[(u;)~(~,~
+ ad + (u;)‘(P13
+ Sdl) + (k 8 k){@,+ PI>.
+i
Next, we compute the normal stress differences
T,, -
G2
=
-[
+
w2(Q12
$ E2Wl,~
+
P2)
+
+
E2(%
+
P3)
+
2Ev’(cu,
+
P4)
+
d w,2[
(4)2%
4P,ol+; ~m4)2(~13 + a141+ (4)‘uL + Pdl}
+
CU32S8]
(28)
(29)
Modeling electro-rheological materials
487
and thus both normal stress differences are in general non-zero. The electrical field, however,
does not contribute to the normals stress difference T,, - 7& if the material functions cu, and &,
are constants. However, in general the material coefficients will depend on the electrical field
and thus the normal stress difference T,, - T3Dwill depend on the applied electric field. We also
observe that the electrical field induces normal stress differences even when the particles are
homogeneously dispersed, i.e. u = constant.
It is interesting to observe that even in the absence of flow, inhomogeneous distribution of
the particles (i.e. v #0) gives rise to normal stress difference T,, - Tz2.
Next, the total shear stress T12is given by
=
bf(v, v’, E, 4)lu; + [ps(v, v’, E, u;)]uI,
(30)
where pf and pS represent generalized shear viscosity functions that depend on the electric field
and the manner in which the particles are distributed in addition to the shear rates.
We notice that even when the material is homogeneously dispersed, i.e. u = constant, the
electric field can cause changes in the generalized viscosities pf and pS. Also, in the absence of
an electric field, the distribution of the solid particles can change the generalized viscosity.
When the particles are homogeneously dispersed the viscosities pf and pS have the form
(31)
(32)
where
(33)
PS,Y
= Ps.Y(E~,
(~3).
(34)
Thus, even the simplest case when the material moduli are constant, the electrical field can
cause the material to shear thicken. Moreover, the electrical field enters the expressions for the
generalized viscosities in a non-linear fashion and thus can produce a significant change in the
viscosity for large values of the field.
Henceforth, we shall assume that the carrier fluid is incompressible. Thus, the function (Y,
should be replaced by the indeterminate field -p. In this case the material can undergo only
isochoric motions and hence has to meet
%,Y
=
ey(E2,
div
(u;)),
uf = 0.
In the case of (23),, this condition is met automatically.
We now turn our attention to deriving the balance of linear momentum of each constituent.
It follows from (5), (6) (24) and (25) that
i
{U;CU~+ (v~)~[u;(w,] +
E2(4)a;, + E~[u;(N,,
+ CY,~)]}’- y,(uf-
(Q”%
(~‘)~a~ + E2q + ~Ev’LY‘,+ ___
4
u,) = g=
fi,
+
(vf)2jj2
{uI/%
+
E2&
+
+
Wz[uLS~l
+
E’WBY
2&f/j,
I (“~2~fi
+
; ;
Ev’[u;(B,,
(u;)2(v,)/.3s
(35)
1
(G2E2a,,,
+ i (U;)2(v’)%, +
4
(a13
: &ld)}=2 ,
+ (u;)2v’E
;
K, = const,
+
+
,%2)]}’
t”?f2h+
+
Y,(ut
-
(u:)2v’EfL
us)
=
(37)
o
+
(36)
h4)]’
=
o.
(38)
K. R.
488
RAJAGOPAL
et al.
The assignment of boundary conditions is not usually straightforward in mixture theory. If
traction boundary conditions are involved, then we are faced with the daunting task of
prescribing individual partial tractions, when in most problems we are only aware of the total
traction. A method for overcoming this difficulty within the context of some special mixtures
has been provided by Rajagopal et al. [12].t If we are studying problems involving velocity
boundary conditions, we would have no difficulty if we assume both the constituents satisfy the
adherence condition. However, it is not clear that the particulate material has to adhere to the
boundary and even in this case we might have an indeterminacy in the problem. In fact, even in
the case of a purely granular solid the nature of the boundary condition is a thorny issue and
far from being settled. Here, we shall assume that both the constituents adhere to the
boundary. Thus (cf. Fig. 1)
u,(O) = 0,
u,(h) = 0,
(39)
&(O) = 0,
u,(h) = 0.
(40)
Thus, we need to solve (35) and (36) subject to (39) and (40). As we mentioned earlier this
would have to be done numerically, and even in the absence of the electrical field this is a
reasonably tedious calculation (cf. Johnson et al. [5, 61).
To make the theory have some practical utility it is necessary to make reasonable
simplifications which retain the important physical effects, but within a structure with far fewer
material parameters. With this in mind, we turn our attention towards a re-evaluation of the
basic assumption on the forms of the constitutive equations ‘I? and T’. If we are interested in
slow flows of electro-rheological
materials in which the volume distribution of the particles is
more or less uniform in the sense that gradients of u are small, we can assume as a starting
point that T” depends linearly on D” and U. As we are interested in problems where E is large,
we cannot ignore the quadratic terms in E. Since the carrier fluid is Newtonian and
non-conducting, and if we furthermore assume that the stress T’ does not depend on 21, the
effect of the particulate distributions manifesting themselves through the interaction terms, the
expressions for the partial solid stress T” and the fluid stress T’ take the form:
T”=~,l+~ZU~U++PoE~E+~~(U~E+EEU)+~sDr+~,(DSE~E+EE~E)
+ P,,(D”U 8 E + E @ DW) + &(DSE
03 U + U 60 D”E),
(41)
and
T’=-~~+~Y~E@E++~D~+~~~(D~E@E+EED~E).
(42)
We now consider an example which illustrates the complexity involved in the simplest of
boundary value problems.
The assumption of linearity in D” and U for instance allows the material coefficients to
depend on the tr D”, and since there are no restrictions on the dependence of E, they can also
depend on the various invariants which depend on E, and also non-linearly on v. We shall
make the further assumption that all the material moduli are constants, except for /I,. The
coefficient /3, plays the role of pressure in a compressible material and thus depends on V. If we
want an equation of state similar to an ideal gas we would pick p, - KV (cf. Rajagopal and
Massoudi [13]). There is also some information on the manner in which these material
coefficients depend on v for pure granular materials. However, we shall not get into a
discussion of these issues here as they are not central to our illustrations.
Since we shall be interested in slow flows, the relative acceleration effect ,Sf can be ignored
and the interactive force simplifies to
b = y,(u” - u’) = y,i(u” - u’).
(43)
_
tFor problems involving non-linearly elastic solids infused with fluids, at saturation, Rajagopal et al. [ 121 require that
the variation in the Helmholtz
free energy equal the work done by the external tractions. This thermodynamic
criterion provides an additional condition which is used in place of a boundary condition.
Modeling electro-rheological materials
489
The study of Johnson et al. [6,7], as we mentioned earlier, considers the problem when E = 0.
Here, we shall consider the other extreme case when V’ =0 (i.e. the material is
homogeneously dispersed). In this case, we obtain
;t
a5u; + E2a9(uj)}'
- y,(uf - u,) =
$ =K,,
U&4 + E*B&l))’ + Y,(W - us)= 0.
;
(46)
When E and u are constant, (38) is automatically satisfied. We thus notice that the flow is due
to the pressure gradient in the carrier fluid, the solid particles moving by virtue of the
interaction forces due to the fluid on the solid particles.
On adding (44) and (45) it immediately follows that equations (44) and (46) can be expressed
as
(P‘G’
- Y&t-
(P&l)
US)= k*,
(47)
+ Y&f - US)= 0.
(48)
From (48) we get
_Ir,u:+us
U‘=
Yi
IV
-E!%+u:*
ul;=
YI
Substituting (49) and (50) in equation (47) we obtain
u,v_~d~f+~s)u”
s
PfPs
:
s
k,y_0
’
Pfcrs
k,<O,
(51)
which implies that
Cl
u,=~e
my
+
_$ e-mo
by*
+
ml
2(Pf
+ C4Y + cs.
PSI
+
Using (49) and (52) we obtain
UfZ
--
ccsc,emlY+ c2eemlY + 4
bf +A)
Yl t
I
+CIeWY
4
+se-mly+
m:
k’y2
2(Pf
+
+c4y+Cs
(53)
I4
where
m: = YI(Pf + I%)
PfPs
The mixture velocity u, and mixture viscosity p,,, are defined through
u,=
PSUs + P’Uf
P”
pm=ps+pf
Cltu; + P&l
Al=
u,
m
(55)
(56)
(57)
K. R.
490
RAJAGOPAL
et al.
where pm, p”, of are the densities of the mixture solid and fluid respectively. In defining the
mixture viscosity, we have used the fact that the shear stress in the mixture is the sum of the
partial shear stresses. Moreover, since we have assumed that the volume fractions are constant
they do not appear in the expression for the mixture viscosity. However, if volume fractions are
allowed to vary, then even the component viscosities pI and p2 would depend on the local
volume fraction of the constituent and are not necessarily constant.
When electro-rheological
fluids are sheared, in the presence of an electrical field, they
exhibit a Bingham fluid-like behavior in that they flow only after a yield stress [u,,(E)] is
reached.
In order to determine the constants cl, c2, c4 and cs, we consider the steady flow of a
mixture between two flat plates with the top plate being stationary [Fig. l(a)] and the top plate
moving (Fig. l(b)], the bottom plate being always held fixed.
For the electro-rheological fluid to flow the shear stress should be greater than or equal to
the yield stress of the mixture.
Thus, for flow to take place we need
Pnl4il-’ 1 Do
(58)
and from equation (57) at y *
Pf4 ly’+ w: (y*= %.
On substituting for u; and ui at y* we get
-wfw
Cl
YI
{
clemlY* - c2emmlY*}+ -
(pf + p,)emlY* -
ml
2
(pf + ps)e-*lY* +
C4(Pf
+
CL,)
+ kl Y * = %
(59)
Equation (59) is evaluated numerically to obtain the value of y*.
Case 1. Poiseuillepow
(cf. Fig. l(a)]
We assume that the solid and fluid adhere to the boundary and that their profiles are
symmetric about the mid-plane as gravity effects are neglected. Thus,
u,( fh) = 0,
(60),
Uf(fh) = 0,
(60)2
u,(+y) = 4(-Y),
(60)7
&(+Y) = u,(-Y).
(W4
Equation (60), implies that (52) has to be an even function,?
c, =c2
and
c4= 0.
and thus
(61)
Using the adherence boundary conditions (60),,* we get
” = (pf
-k,
+ ps)(emIh+ eemlh)
(62)
and
(63)
ffn a real problem it is possible that the electric field varies in the flow region, which would imply a flow field that is not
necessarily symmetric about a mid-plane. In fact, in order to study the problem fully, we would have to solve for
the electric field. However,
here, we are using the electric field like a fixed parameter that enters the problems.
Moreover,
we note from (47) and (48) that all that is necessary for determining
the solutions are boundary
conditions (60), and (60),. The assumptions (60)s and (60), help in greatly simplifying the method of solution, and
is not inconsistent with the other assumptions and physical expectation.
Modeling electro-rheological
materials
491
ThUS
u,
G
=
(emlY + eemlY)
hy*
+
ar+
ml
Ps
uf=--
Yl
Case II. Couette
+c
PSI
(64)
s
I(
c, emly+e
(65)
flow[cf. Fig. l(b)]
We assume that the top plate moves with a speed U along the x-direction and that the fluid
and granular solid adhere to the boundary. Then the boundary conditions are
u,(h) = u,
(66) I
U,(Y *) = 0,
(66)2
u,(h) = I/,
(6%
uxy*) = 0,
(6%
where y* denotes the y-coordinate at which flow is initiated.
It follows from (66),, (52) and (53) that
(e-mlh
k,
c2= (pt
_
(e-“‘Y’
+ p,)
-
kle-""Y*
cl = -(pf
5
(emly*
_ em~h) + c2
+ ps)
(e-m~y’
-
c2e
_ e-mfih)
e-wY')
-2m,y*
+
4
cs = _A_
emlY*
_
m:
3
e-m’Y’
_
(67)
,-w”)
(68)
2(1”k; pS) (Y *2 - h’)]/(h
-Y *) (69)
by*’
(70)
2(/&f+ /&) - c4y **
ml
Thus
+ 2c2e-“‘y + by2
&Le,lY
mf
Uf = 3
ml
k,
C,emlY
YI
I
2(Pf
+ C2e-m’Y
+ (pf
+ ps)
+ I%)
+ C4Y
+ cs
+zlewy
I
(71)
+
m:
c4y
+
cg.
(72)
The mixture velocity (u,) and mixture viscosity (p,,,) are computed using (55) and (57),
respectively.
Table 1 shows the values of a,,, k,, yI , ps, pf, p”, pf used to generate the velocity profiles for
Poiseuille flow and Table 2 shows the values of a,,, k,, u, yI, ps, pf, p’, pf used in Couette flow.
Table 1
00
k,
(Pa)
(Pa/m)
8
-64
4
2
0
-64
-64
-64
(k:is) (kg%s)
0.1
1
IO
1
1
1
Note: pS = 1.5 pf, p’= 3.5 p’.
0.1334
891.2
0.125
0.1334
0.1334
0.1334
891.2
891.2
891.2
0.0625
0.03125
0
K. R. RAJAGOPAL
492
et al.
Table 2
%
(Pa)
k,
(Pa/m)
&s)
u
(m/s)
Pf
(Wms)
(kg$)
&
16
-64
0.1
1
10
0.1334
891.2
10
0.34169
8
4
4
4
-64
-64
-64
-64
I
1
0.1334
0.1334
0.1334
0.1334
891.2
891.2
891.2
891.2
IO
10
100
10
0.13018
0.01879
0.0
0.11325
1
1
Note: pc, = 1.5 pf, pS = 3.5 pf.
In the case of Poiseuille flow, Figs 2-4 illustrate the effect of increasing the electric field,
which results in an increase in the yield stress. As expected, increasing the yield stress increases
the plug region. Figure 2 corresponds to the classical Poiseuille flow in the absence of the
electric field. The velocity of the solid, fluid and mixture have been normalized with respect to
the maximum fluid velocity. Figures 4, 5 and 6 show the effect on the velocity profiles due to a
variation in the interaction parameter yI which is akin to Stokes drag on a particle. It may be
noticed that changes in yI neither enhance nor diminish the plug flow domain, but as we would
expect it affects the difference in velocities between the solid and the fluid. As y,, increases the
difference between the speeds of the fluid and solid increases. Notice that the mixture velocity
always lies within those of the solid and fluid.
In the case of the flow between flat plates with the top plate moving, and in the presence of a
pressure gradient, we observe that an increase in the yield stress (a,,) increases the plug region
/.hf’0.1334 Kg/ms
p’=891.2
Kg/m3
a,=0
~1’1
Pa
2H=l m
Kg/s
Kl=-64
Pa/m
‘x.,
.....
‘...
‘..,
‘...
,.:
.:’
,:’
,:’
.:’
,./
,....’
0.00
0.25
Fig. 2. Normalized
velocity
vs normalized
1.00
0.75
0.50
NORMALIZED
VELOCITY
distance
(Poiseuille
flow-no
electric
field).
Modeling electro-rheological
jq’O.1334
ps=1.5l-q
K g / ms
p’=891.2
p’=3.5p’
Kg/m3
a,=4
materials
yl=l
Pa
Kg/s
Kl=-64
493
ZH=l m
Pa/m
1
MIXTURE
.-___________e.
4
I
0.25
0.00
0.50
0.75
1.00
NORMALIZED VELOCITY
Fig. 3. Normalized velocity vs normalized distance (Poiseuille flow-with
yf=O.1334
1-Ls=l.5l-q
Lo
Kg/ms
p’=891.2
p’=3.!$
Kg/m’
a,=8
yl=l
Pa
Kg/s
Kl=-64
electric field).
2H=l m
Pa/m
SOLID
il.00
0:25
0.50
I
I
0.75
1.00
NORMALIZED VELOCITY
Fig. 4. Normalized velocity vs normalized distance (Poiseuille flow).
K. R. RAJAGOPAL et al.
494
pf=0.1334
M
/+=1.5bq
K g /ms p’=891.2
pY=3.5p’
Kg/m3
yl=O.i
u0=8 Pa
Kg/s
Kl=--64
ZH=l m
Pa/m
Fig. 5. Normalizedvelocity vs normalizeddistance(Poiseuille flow).
/..q=Q.1334
v!
0
/+=mq
K g /ms p’=891.2
p*=3.5pr
Kg/m3 yl=iO
(r,=8
Pa
Kg/s
K1=-64
ZH=l m
Pa/m
SOLID
FLUID
..,....
.,.1.. . . I... .
MIXTURE
_____.._w___*_.
I
1.00
/
NORMALIZED VELOCITY
Fig. 6. Normalizedvelocity vs normalizeddistance(Poiseuilleflow).
Modeling electro-rheological materials
Kg/ms p'=891.2Kg/m' yl=lKg/s
pf=O.1334
0
u,=4 Pa
rus'l.5clf p'=3.5pr
_ .:
z o-
,’
0.0
0.1
495
0.2
0.3
I
0.4
Kl=-64 Pa/m
,
...”
I
0.5
H=lm
0.6
I
0.7
U=lO m/s
..’
I
0.6
SOLID
I
0.9
1.0
NORMALIZEDVELOCITY
Fig. 7. Normalized velocity vs normalized distance (Couette flow).
0
/q=O.1334Kg/ms p'=891.2Kg/m3 yl=lKg/s
H=lm
uo=8 Pa
/-$=1.5kq p'=3.5p'
U=lO m/s
Kl=-64 Pa/m
SOLID
0.0
0.1
0.2
/
1
I
I
0..3
0.4
0.5
0.6
1
0.7
0.R
0.9
1.0
NORMALIZEDVELOCITY
Fig. 8. Normalized velocity vs normalized distance (Couette flow).
ii. R. RAJAGOPAL
4%
pf=O.1334
Kg/ms pf=89L.2 Kg/m3
p*=d.$’
/ls=wq
oo=16 Pa
et a/.
yl=l
Kl=-64
Kg/s
H=l m
Pa/m. U=lOm/s
MlXTURE
x
0.0
0.1
0.2
0.3
0.4
03
0.6
0.7
0.8
03
1.0
NORMALIZED VELOCITY
Fig. 9. Normalized velocity vs normalized distance {Couette flow).
,U@.1334
Kg/ms
p’=891.2
p”d.5p’
cLs’l.fbf
9
..I-
Kg/m3
no=16 Pa
yl=O.i
Kl=-64
Kg/s
Pa/m
H=l m
U=lO m/s
‘...,,,
“%,_,..*
***....
**.,
.I,.1%.
‘...
1...
XI
o-
‘,.,
SOLID
.m
d-
MIXTURE
*-.I**__._**_..
0
da
0.0
ii.1
I
0.2
/
0.3
0.4
I
0.5
/
0.6
0.7
0.8
/
0.9
I
1.0
NORMALIZED VELOCITY
Fig. 10. Normalized velocity vs normalized distance (Chette
flow).
Modeling electro-rheological materials
jq=O.1334
0.0
0.1
497
Kg/ms p'=89l.EKg/m3 yl=lO Kg/s
0.2
0.3
0.4
0.5
0.6
0.7
0.9
H=lm
0.9
1.0
NORMALIZEDVELOCITY
Fig. 11. Normalized velocity vs normalized distance (Couette flow).
pf=0.1334
H=lm
Kg/ms p'=891.2Kg/m3 yl=l Kg/s
Kl=-64 Pa/m
a,=4 Pa
/$=1.5/-q p'=3.5p'
0
_;-
U=lm/s
m
d"....
.......
'...
....
....
...*
w'-_
u"
&_
rno
0"
P"_
wN"
2
&ZI
&
Zd-
..:
:.'
,...
........
.
.
.
.
.......'
SOLID
MIXTURE
-.__--w.__1-...
d0
d
0.0
0.1
I
0.2
I
0.3
I
0.4
0.5
,
0.6
,
0.7
I
0.8
I
0.9
I
1.0
NORMALIZEDVELOCITY
Fig. 12. Normalized velocity vs normalized distance (Couette flow).
ES 32:3-H
498
K. R.
/q=O.1334
0
K g / ms
pr=891.2
p’=3.5p’
k$=1.5bq
et al.
RAJAGOPAL
Kg/m3
u,=4
Pa
yl=l
Kg/s
Kl=-64
H=l
Pa/m
m
U=lOO
m/s
SOLID
I
0.1
0.0
0.2
0.3
0.4
NORMALIZED
Fig. 13. Normalized
wf’O.1334
cLs’l.5bq
,
0.5
d.6
-
0.7
0.9
I.0
VELOCITY
velocity vs normalized
Kg/ms
0.8
p’=891.2
p’=3.5p’
distance (Couette
Kg/m3
u,=8
yl=l
Kg/s
Pa
Kl=-64
2,
flow).
ZH=l
m
Pa/m
,
*’
#’
?
o-
8’
~
,’
,’
Fig. 14. Normalized
velocity vs normalized
distance between
plates (Poiseuille
flow).
Modeling electro-rheological
pf=0.1334
Kg/ms p’=891.2
p’=3.5p’
6 I
0.0
/
,
0.1
0.2
materials
Kg/m3 yl=l
(r,=16 Pa
I
0.3
Kl=-64
I
0.4
MIXTURE VISCOSITY Kg/ms
499
Kg/s
Pa/m
H=l m
U=lO m/s
,
0.5
1
0.6
Fig. 15. Mixture velocity vs distance between plates (Couette flow).
(Figs 7-9), similar to that observed in the case of Poiseuille flow. However, unlike the problem
of Poiseuille flow the plug region is not symmetrically placed with respect to the flow field, but
is adjacent to the fixed bottom plate. Of course, increasing the pressure gradient and the plate
velocity such that the shear stress is larger than the yield stress would eliminate the plug region
altogether. On increasing the interaction parameter y,, the region of zero velocity does not
change (Figs 9-ll), but the difference between normalized velocities of the solid and fluid
decreases. Figures 7, 12 and 13 illustrate the effect of changing the velocity of the top plate.
Increasing the velocity of the top plate decreases the region of zero velocity (plug region) and
this is to be expected as the shear stresses in the flow domain is higher than in the case with
lower top plate velocity.
In both the cases considered, the solid and fluid viscosities are taken as constants, but as
illustrated in Figs 14 and 15, it is observed that the mixture viscosity as defined through (57)
changes with position. The fact that the difference between the fluid and solid velocities and
their respective velocity gradients change with position induces the mixture viscosity as defined
through (57) to change with position.
The above solutions are in complete agreement with the predictions of the homogenized
single continuum theory of Rajagopal and Wineman [2] in that there is a central region in
which there is plug flow, for the mixture.
Acknowledgement-K.
R. Rajagopal thanks the Air Force Office of Scientific Research for its support.
K. R. RAJAGOPAL
500
et al.
REFERENCES
[1]
[2)
[3]
[4]
[S]
[6]
[7]
[8]
[9]
[lOI
. -
W. M. WINSLOW, J. Appl. Phys. 20, 1137 (1949).
K. R. RAJAGOPAL and A. S. WINEMAN, Acfa Mech. 91, 57 (1992).
C. TRUESDELL, Rend. Lincei Ser 8 22, 33 (1957).
C. TRUESDELL, Rational Thermodynamics,
2nd edn. Springer, Berlin (1984).
M. MASSOUDI, Int. J. Engng Sci 26, 765 (1988).
G. JOHNSON, M. MASSOUDI and K. R. RAJAGOPAL, Chem. Engng Sci. 46, 1713 (1991).
G. JOHNSON, M. MASSOUDI and K. R. RAJAGOPAL, Inr. J. Engng Sci. 29, 649 (1991).
J. E. ADKINS, Phil. Trans. R. Sot. 22SA, 607 (1963).
C. TRUESDELL, A First Course in Rational Confinuum Mechanics, Vol. 1. Academic Press, New York.
A. J. M. SPENCER, Theory of invariants. In Confinuum Phvsics
(Edited bv A. C. ERINGEN). / Academic Press.
_
_
New York (1974).
[11] G. JOHNSON, M. MASSOUDI and K. R. RAJAGOPAL, A review of interaction mechanisms in fluid-solid
flows. Topical Report, DOE/PETC-TR-90-9 (1990).
[12] K. R. RAJAGOPAL, A. S. WINEMAN and-M. GANDHI, Int. J. Engng Sci. 24, 1453 (1986).
[13] K. R. RAJAGOPAL and M. MASSOUDI, A method for measuring the material moduli of granular materials:
flow in an orthogonal rheometer. Topical Report, DOE/PETC/TR-90/3.
(Received
2 February
1993; accepted
14 April
1993)